Finite Group Databasebeta

← #6434682

Subgroups — #6434682

Order 512.

3140 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic, complemented
2C2×#3094C211yesmaximal, characteristic
3C22×#480C211yesmaximal, characteristic
4#35262C211yesmaximal, characteristic
5#29646C218yesmaximal, complemented
6#29640C214yesmaximal, complemented
7#29638C214yesmaximal, complemented
8#23555C214yesmaximal, complemented
9#23456C214yesmaximal, complemented
10#20700C211yesmaximal, characteristic, complemented
11#20699C211yesmaximal, characteristic, complemented
12#20685C211yesmaximal, characteristic, complemented
13C2×#544—22no—
14C2×#544—22no—
15C2×#544—28no—
16C2×#544—28no—
17C2×#544C2214yes—
18C2×#544C2214yescomplemented
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2212yescomplemented
21C22×#125—21no—
22C22×#125C2212yescomplemented
23C22×C22⋊Q8—22no—
24C22×#121C2211yescharacteristic
25C22×#117C2211yescharacteristic
26C22×#117C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28#3094—28no—
29C2×#481C2212yes—
30C2×#482C2214yes—
31C2×#480—28no—
32C2×#480C2214yes—
33C2×#480C2214yes—
34C2×#480C2212yes—
35C2×#428C2212yes—
36C22×#107C2212yes—
37C2×#418C2212yes—
38C22×#102C2212yes—
39C2×#407C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41#2854C2214yes—
42#2624C2214yes—
43#2504—21no—
44#2501—21no—
45#2500—22no—
46#2482—22no—
47#2405—22no—
48#2317C2212yes—
49#2309C2212yes—
50#2280C2211yescharacteristic
51#2059C22116yescomplemented
52#2057C22116yescomplemented
53#2025C22116yescomplemented
54#2023C22116yescomplemented
55#1825C2214yescomplemented
56#1824C2212yescomplemented
57#1824C2214yescomplemented
58#1823C2212yescomplemented
59#1823C2214yescomplemented
60#1818C2214yescomplemented
61#1817C2214yescomplemented
62#1736C2214yes—
63#1735C2214yes—
64#1729C2214yes—
65C23×Q8—22no—
66C23×D4—21no—
67C23×D4C2311yescharacteristic
68C24×C4—21no—
69C24×C4C2311yescharacteristic
70C22×M16C2312yes—
71C23×C8C2312yes—
72#544—232no—
73#544—232no—
74#544—216no—
75#544—216no—
76#544—216no—
77#544—216no—
78C2×C4⋊Q8—216no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8C2318yes—
85C2×#125—216no—
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—28no—
89C2×#125—28no—
90C2×#125—24no—
91C2×#125C2318yes—
92C2×#123—24no—
93C2×C22⋊Q8—232no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×#121—21no—
97C2×#121—21no—
98C2×#121—24no—
99C2×#121—24no—
100C2×#121C2312yes—
101C2×#121C2312yes—
102C2×#117—24no—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117D418yes—
108C2×#117D412yes—
109C2×#117D412yes—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D418yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D412yes—
134C22×C42—21no—
135C22×C42C2312yes—
136#482—216no—
137#481—28no—
138#480—216no—
139#480—216no—
140#480—28no—
141#428—28no—
142C2×#107—28no—
143C2×#107C2314yes—
144C2×#107C2318yes—
145C2×#106C2314yes—
146#418—28no—
147C2×#102—28no—
148C2×#102C2318yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151#407—24no—
152C2×C22⋊C8—24no—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157#399—24no—
158#397—24no—
159#394—28no—
160#390—22no—
161#390—22no—
162#390C2318yes—
163#389C2318yes—
164#388—28no—
165#386—24no—
166#385—24no—
167#384—22no—
168#382—22no—
169#381—21no—
170#380C2312yes—
171#380C2312yes—
172#379—21no—
173#379—24no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176#340C2318yes—
177#340C23116yes—
178#340C2318yes—
179#336C23116yes—
180C25—21no—
181C25C2×D411yescharacteristic
182C22×Q8—216no—
183C22×Q8—28no—
184C22×Q8—88no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—21no—
190C22×D4—28no—
191C22×D4—44no—
192C22×D4—84no—
193C22×D4C2412yes—
194C23×C4—28no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—84no—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C2×M16—24no—
214C2×M16—28no—
215C22×C8—24no—
216C22×C8—28no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—232no—
220C4⋊Q8—232no—
221C4⋊Q8—232no—
222C4⋊Q8—216no—
223C4⋊Q8—216no—
224#125—232no—
225#125—232no—
226#125—232no—
227#125—216no—
228#125—216no—
229#123—88no—
230C22⋊Q8—264no—
231C22⋊Q8—264no—
232#121—24no—
233#121—24no—
234#121—24no—
235#121—24no—
236#121—44no—
237#121—44no—
238#117—28no—
239#117—28no—
240#117—28no—
241#117—28no—
242#117—24no—
243#117—24no—
244#117—24no—
245#117—24no—
246#117—24no—
247#117—24no—
248#117—28no—
249#117C2×D418yes—
250#117C2×D418yes—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42C2×D414yes—
311C2×C42C2414yes—
312#107—216no—
313#107—28no—
314#106—28no—
315#102—216no—
316#102—28no—
317#102—28no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322#95—22no—
323#95—22no—
324#95—28no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—42no—
329C24—42no—
330C24—44no—
331C24—82no—
332C24—82no—
333C24#12014yes—
334C24C22×D411yescharacteristic
335C24C22×D411yescharacteristic
336C24C22×D411yescharacteristic
337C2×Q8—232no—
338C2×Q8—816no—
339C2×Q8—832no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—28no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4—42no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—84no—
351C2×D4—816no—
352C22×C4—24no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—816no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—82no—
381C22×C4#12014yes—
382C22×C4#12014yes—
383C22×C4#12014yes—
384C22×C4C22×D414yes—
385C22×C4C22×D414yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C42−1+418yes—
396C22×C4C2511yescharacteristic
397M16—48no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—28no—
401C2×C8—28no—
402C2×C8—48no—
403C4⋊C4—232no—
404C4⋊C4—232no—
405C4⋊C4—232no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—28no—
421C4⋊C4—42no—
422C4⋊C4—88no—
423C4⋊C4C22×D414yes—
424C4⋊C4C22×D418yes—
425C4⋊C4C22×D414yes—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—82no—
437C42—28no—
438C42—28no—
439C42—28no—
440C42—24no—
441C42—24no—
442C42—44no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—28no—
449C23—22no—
450C23—28no—
451C23—22no—
452C23—44no—
453C23—416no—
454C23—44no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—84no—
459C23—84no—
460C23—84no—
461C23—168no—
462C23C2×#12012yes—
463C23C2×#12012yes—
464C23C2×#12012yes—
465C23#53614yes—
466C23#57614yes—
467C23C23×D411yescharacteristic
468Q8—832no—
469D4—44no—
470D4—44no—
471D4—88no—
472D4—88no—
473D4—84no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—416no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—88no—
504C2×C4—88no—
505C2×C4—84no—
506C2×C4C2×#12012yes—
507C2×C4C2×#12012yes—
508C2×C4C2×#12012yes—
509C2×C4#53614yes—
510C2×C4#53618yes—
511C2×C4#53618yes—
512C2×C4C23×D411yescharacteristic
513C2×C4C23×D411yescharacteristic
514C2×C4C2×2−1+414yes—
515C8—44no—
516C8—44no—
517C22—22no—
518C22—28no—
519C22—28no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—416no—
527C22—44no—
528C22—82no—
529C22—88no—
530C22—82no—
531C22—82no—
532C22—168no—
533C22#261414yes—
534C22#300814yes—
535C22C22×#12011yescharacteristic
536C22C2×#53612yes—
537C22C2×#57612yes—
538C22#358612yes—
539C4—22no—
540C4—22no—
541C4—44no—
542C4—48no—
543C4—44no—
544C4—44no—
545C4—44no—
546C4—84no—
547C4—82no—
548C4#261414yes—
549C4C2×#53612yes—
550C4C2×#53612yes—
551C2—24no—
552C2—24no—
553C2—42no—
554C2—162no—
555C2C2×#261411yescharacteristic
556C2C2×#300811yescharacteristic
557C2#6036811yescharacteristic
558C2#6087114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#544C2semidirect
5. #29646C2semidirect
6. #29640C2semidirect
7. #29638C2semidirect
8. #23555C2semidirect
9. #23456C2semidirect
10. #20700C2semidirect
11. #20699C2semidirect
12. #20685C2semidirect
18. C2×#544C22semidirect
20. C22×C4⋊Q8C22semidirect
22. C22×#125C22semidirect
51. #2059C22semidirect
52. #2057C22semidirect
53. #2025C22semidirect
54. #2023C22semidirect
55. #1825C22semidirect
56. #1824C22semidirect
57. #1824C22semidirect
58. #1823C22semidirect
59. #1823C22semidirect
60. #1818C22semidirect
61. #1817C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#5442
C2×#30942
C22×#4802
#352622
#296462
#296402
#296382
#235552
#234562
#207002
#206992
#206852